Explanation

(a)
Let $I = \int_{0}^{\pi/2}{\frac{\sin^{2}x}{\sin^{2}{xcos^{2}x}}dx}$ …(i)
$\Rightarrow I = \int_{0}^{\pi/2}{\frac{\cos^{2}x}{\sin^{2}{x + \cos^{2}x}}dx\ \ \ \ \ \ldots(ii)}$
On adding Eqs. (i) and(ii), we get
$2I = \int_{0}^{\pi/2}{1\ dx = \lbrack x\left. \ \ \right\rbrack_{0}^{\pi/2}} = \frac{\pi}{2}$
$\Rightarrow \ I = \frac{\pi}{4}$